The theory of determinants in the historical order of development, by Sir Thomas Muir.

CIRCULANTS (BALTZER, 1864)

375

He does not, however, note, as Beltrami did in a special case early
in the same year, that by using another method of elimination,
namely, Euler's of 1748, the eliminant is found to be
r=l
II(ag-y + alOr + a2r +... + an10'),'
where 0r is an nth root of 1: and thus he fails to bring out the fact
that Spottiswoode's result of 1853 is nothing more or less than the
statement of the equality of those two eliminants.* Instead of this
he performs the operation
coli + Ocol2 + 0,cols +.. *- + -'ol,
and so arrives at a practically equivalent result, namely, that the
equation
C(ao-y, a,, a,.., a,_) = 0
is satisfied by putting
y = a0 + al r + a20r +... + an-_O1.
Two other points worth noting are (1) his calling
C(ao, al, a2,..., an_)
the norvm of
ao + alOr + a2O +.. + a. _- 1,
in accordance with an extension of a usage of Gauss', and (2) his
statement that of the n" terms got by working out the product
( (ao + aiOr + a20 +... + a on1)
only the 1 2 3.... n terms of the determinant remain. In
regard to the former it has to be remarked that 0r must then be
restricted to stand for a primitive ntl root of 1, and in regard to the
latter that even some of the 1 ~ 2 * 3... n terms of the determinant
do not remain.
WOLSTENHOLME, J. (1867).
[A BOOK OF MATHEMATICAL PROBLEMS,.... 344 pp. London.]
Result no. 920 is
C(cos 0, cos 2, cosn0) -= cos 0-cos(n+-1)0}-{1 —osn
2 (1 - cos nO)
after deleting a sign-factor.
* See Hist., ii. pp. 369-370.